Generalizations of Samuel's criteria for a ring to be a unique factorization domain

Actions of SL2(k) on Affine k-Domains and Fundamental Pairs

“…By Corollary 4.2, T 0 , T 1 , T 2 are irreducible and pairwise relatively prime in A. Since A is factorially closed in B, T 0 , T 1 , T 2 are irreducible and pairwise relatively prime in B = k [4] . By Lemma 6.3, β 2 = 1, which gives a contradiction.…”

“…Corollary 7.3. Let B = k [4] with an effective Z 2 -grading, let D ∈ LND(B) be nonzero and homogeneous, and let A = ker D. If A 0 = k then either A ∼ = k [3] or…”

“…, x m and for D. In this way, one obtains an effective unmixed Z m+1 -grading of B with B 0 = k for which D is homogeneous. In case n = 4, this gives a Z 2 -grading and Corollary 7.3 shows that either ker D ∼ = k [3] or ker D ∼ = k[x, y, u, v]/(x a + y b + u c v d ) for certain integers a, b, c, d. In [12], Maubach proved by another method (without assuming k is algebraically closed) that a triangular monomial derivation of k [4] has kernel isomorphic to k [3] or k[x, y, u, v]/(x a + y b + u c v), i.e., d = 1.…”

“…Theorem 1.1. ( [4], Theorem 5.1) Let k be any field. Given trinomial data ∆ over k, k[∆] is an affine rational UFD of dimension n − r + 1 ≥ 2 over k.…”